Chapter 6

Find the point of the plane \(2 \mathrm{x}-3 \mathrm{y}-4 \mathrm{z}=25\) which is nearest the point \((3,2,1)\).

For the following quadratic forms, tell by inspection whether the origin is a maximum or minimum: \(\mathrm{q}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+\mathrm{y}^{2}\) \(q(x, y)=x^{2}-y^{2}\) \(q(x, y)=x y\)

Let the function \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) be continuous and have continuous first and second partial derivatives in a region \(R .\) Let \(\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)\) be an interior point of \(\mathrm{R}\) for which ( \(\left.\partial \mathrm{f} / \partial \mathrm{x}\right)=0\) \((\partial \mathrm{f} / \partial \mathrm{y})=0 .\) Given the condition \(\left[\mathrm{f}_{12}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)\right]^{2}-\mathrm{f}_{11}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)\) \(\mathrm{f}_{22}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)<0\) and \(\mathrm{f}_{11}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)<0\) prove that \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) has a relative maximum at \(\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)\).

Find the critical points of the function \(\mathrm{f}(\mathrm{x}, \mathrm{y})=4 \mathrm{xy}-2 \mathrm{x}^{2}-\mathrm{y}^{4}\). Then determine whether each is a relative maximum, relative minimum or a saddle point of \(\mathrm{f}(\mathrm{x}, \mathrm{y})\).

Let \(\mathrm{f}\) be the function given by \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+\mathrm{y}^{3}-3 \mathrm{xy}\). Find the critical points of \(\mathrm{f}(\mathrm{x}, \mathrm{y})\). Then determine whether each critical point is a relative maximum, relative minimum or saddle point of \(\mathrm{f}(\mathrm{x}, \mathrm{y})\).

a) Find whether the origin is a relative maximum or minimum, or neither for the function \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\log \left(1+\mathrm{x}^{2}+\mathrm{y}^{2}\right)\). b) Find the critical points of the function $$ f(x, y)=x^{2}-12 y^{2}+4 y^{3}+3 y^{4} $$ and determine whether each is a relative maximum or saddle point of \(\mathrm{f}(\mathrm{x}, \mathrm{y})\)

Find the critical points and the nature of each critical point (i.e., relative maximum, relative minimum, or saddle point) for: a) \(f(x, y)=x^{2}-2 x y+2 y^{2}+x-5\) b) \(\mathrm{f}(\mathrm{x}, \mathrm{y})=(1-\mathrm{x})(1-\mathrm{y})(\mathrm{x}+\mathrm{y}-1)\).

Find the critical points of the function $$ f(x, y)=x^{4}+y^{4}-x^{2}-y^{2}+1 $$ Then determine if each critical point is a relative maximum, relative minimum, or saddle point.

Find the values of \((x, y, z)\) that minimize $$ F(x, y, z)=x y+2 y z+2 x z $$ given the condition \(\mathrm{G}(\mathrm{x}, \mathrm{y}, z)=\mathrm{xyz}=32\).

Find the maximum and minimum values of \(\mathrm{F}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\) on the surface of the ellipsoid \(\mathrm{G}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\left(\mathrm{x}^{2} / 64\right)+\left(\mathrm{y}^{2} / 36\right)+\left(\mathrm{z}^{2} / 25\right)=1\)

Find the maximum value of \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy}\); \((\mathrm{xy}>0)\) subject to the constraint \(\mathrm{x}^{2}+\mathrm{y}^{2}=8\) by drawing the level curves and by another method.

For the following quadratic form $$ \mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{ax}^{2}+2 \mathrm{bxy}+\mathrm{cy}^{2} $$ state conditions for \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) to have a minimum and maximum value, using eigenvalues and the side condition \(\mathrm{x}^{2}+\mathrm{y}^{2}=1\).

Find the maximum and minimum of \(z=x^{2}+2 y^{2}-x\) on the set \(x^{2}+y^{2} \leq 1\)

Find the maximum and minimum of \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy}-\mathrm{y}+\mathrm{x}-1\) in the closed disk \(\mathrm{D}=\\{\mathrm{P}:|\mathrm{P}| \leq 2\\}\) ( \(\mathrm{P}\) is a point in the \(\mathrm{xy}\) -plane).

Find the maximum value of the function $$ f(x, y)=4 x y-2 x^{3}-y^{4} $$ in the square \(\mathrm{D}=\\{(\mathrm{x}, \mathrm{y}):|\mathrm{x}| \leq 2,|\mathrm{y}| \leq 2\\}\).

Find the points which might furnish relative maxima and minima of the function $$ \mathrm{f}(\mathrm{x}, \mathrm{y})=2 \mathrm{xy}-\left(1-\mathrm{x}^{2}-\mathrm{y}^{2}\right)^{3 / 2} $$ in the closed region \(\mathrm{x}^{2}+\mathrm{y}^{2} \leq 1\)

Find the extrema for the function \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\) subject to the constraint \(x^{2}+2 y^{2}-z^{2}-1=0\)

Let the number 12 equal the sum of three parts \(\mathrm{x}, \mathrm{y}, \mathrm{z}\). Find values of \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) so that \(\mathrm{xy}^{2} \mathrm{z}^{2}\) shall be a maximum (given the first condition and that \(\mathrm{x}, \mathrm{y}, \mathrm{z}>0)\).

Find the maximum of \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy}\) on the curve $$ G(x, y)=(x+1)^{2}+y^{2}=1 $$ assuming that such a maximum exists.

a) Find the maxima and minima of \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+\mathrm{y}^{2}\) on the ellipse \(\mathrm{G}(\mathrm{x}, \mathrm{y})=2 \mathrm{x}^{2}+3 \mathrm{y}^{2}=1\) b) Find the maxima value of \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{xyz}\) on the plane \((x / a)+(v / b)+(z / c)=1(a, b, c>0)\)

Find the maximum value of \(\mathrm{xyz}\) on the unit sphere \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=1\) by the Lagrange method.

Find the maximum and minimum values of \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\) subject to the constraints \(x^{2} / 4+y^{2} / 5+z^{2} / 25=1\) and \(x+y-z=0\)

Minimize \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+\mathrm{y}^{2}\) subject to \(\mathrm{g}(\mathrm{x}, \mathrm{y})=(\mathrm{x}-1)^{3}-\mathrm{y}^{2}=0\) a) graphically b) using the Lagrangian multiplier method.

\(\mathrm{M}\) where \(\mathrm{p}_{1}=1, \mathrm{p}_{2}=2\) and \(\mathrm{M}=10 .\) Check the second-order conditions to verify that the solution is indeed a maximum.

Let \(\mathrm{f}\) be the following quadratic form \(f(x, y, z)=x^{2}+y^{2}+3^{2}-x y+2 x z+y z\) Find a relative minimum of \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})\).

Find the extremal values of the function \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{xyz}(1-\mathrm{x}-\mathrm{y}-\mathrm{z})\) on the set \(D=\\{(x, y, z) \mid 0 \leq x, 0 \leq y, 0 \leq z, x+y+z \leq 1\\}\)

Find a relative minimum for \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\) $$ 2 x^{2}+2 y^{2}+2 z^{2}-2 x z-2 y z-6 x+2 y+8 z+14 $$

Let A denote a real symmetric matrix of order \(\mathrm{n}\). Find the maximum of \(\mathrm{f}(\underline{\overline{\mathrm{X}}})=\underline{\overline{\mathrm{X}}}^{\mathrm{t}} \mathrm{A} \underline{\overline{\mathrm{X}}}\left(\underline{\overline{\mathrm{X}}}\right.\) is a column matrix, and \(\underline{\mathrm{X}}^{\mathrm{t}}\) is its transpose) on the sphere \(\underline{\bar{x}}^{t} \underline{\bar{x}}=R^{2}(\neq 0)\).

Obtain the Cauchy-Schwartz inequality by extremalizing \(a_{1} x_{1}+\ldots+a_{n} x_{n}\), on the compact surface \(x_{1}^{2}+\ldots+x_{n}^{2}=C\) \(\left(\right.\) Assuming \(\left.a^{\rightarrow} \neq 0\right)\)

Let \(\mathrm{p}, \mathrm{q}\) be positive real numbers such that \(\mathrm{p}^{-1}+\mathrm{q}^{-1}+=1\). Consider the function \(\mathrm{f}=\mathrm{a}^{\rightarrow} \cdot \mathrm{X}^{-}={ }^{\mathrm{n}} \sum_{\mathrm{i}=1} \mathrm{a}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}\), for any \(\mathrm{a}^{\rightarrow}\), with all \(\mathrm{a}_{\mathrm{i}}>0\), on the compact set \(\mathrm{S}=\left\\{\mathrm{X}^{-} \mid\right.\) all \(\left.\mathrm{x}_{\mathrm{i}} \geq 0, \mathrm{x}^{\mathrm{p}}_{\mathrm{i}}+\ldots+\mathrm{x}^{\mathrm{p}}_{\mathrm{n}}=1\right\\}\). Show that the maximum value of \(\mathrm{a}^{\rightarrow} \cdot \mathrm{X}^{-}\) occurs at a point where all \(\mathrm{x}_{\mathrm{i}}>0\). Then, using this method, derive Holder's inequality.

Use Lagrange's method of finding the maximum of functions subject to constraints to develop a) Cauchy's inequality b) Holder's inequality c) Minikowski's inequality.

Let \(\mathrm{f} \in \mathrm{C}^{1}(\mathrm{E})\), the set of all continuously differentiable functions with domain \(E, E\) open in \(R^{n}\), and let \(D_{v} f(p)\) denote the directional derivative in the direction \(\mathrm{v}\) at \(\mathrm{p} \in \mathrm{E}\). What is \(\max _{|| v \mid=1}\left|D_{v} f(p)\right|\) and for which v's of norm 1 is the maximum achieved?

Show that any two hyperplanes in \(\mathrm{R}^{\mathrm{n}}\) either intersect or are parallel by using the method of Lagrange multipliers.